Classification of finite reparametrization symmetry groups in the three-Higgs-doublet model Igor P. Ivanov1;2, Evgeny Vdovin2 1 IFPA, Universit´ede Li`ege,All´eedu 6 Ao^ut17, b^atiment B5a, 4000 Li`ege,Belgium 2 Sobolev Institute of Mathematics, Koptyug avenue 4, 630090, Novosibirsk, Russia February 18, 2013 Abstract Symmetries play a crucial role in electroweak symmetry breaking models with non- minimal Higgs content. Within each class of these models, it is desirable to know which symmetry groups can be implemented via the scalar sector. In N-Higgs-doublet mod- els, this classification problem was solved only for N = 2 doublets. Very recently, we suggested a method to classify all realizable finite symmetry groups of Higgs-family trans- formations in the three-Higgs-doublet model (3HDM). Here, we present this classification in all detail together with an introduction to the theory of solvable groups, which play the key role in our derivation. We also consider generalized-CP symmetries, and dis- cuss the interplay between Higgs-family symmetries and CP -conservation. In particular, we prove that presence of the Z4 symmetry guarantees the explicit CP -conservation of the potential. This work completes classification of finite reparametrization symmetry groups in 3HDM. 1 Introduction The nature of the electroweak symmetry breaking is one of the main puzzles in high-energy physics. Very recently, the CMS and ATLAS collaborations at the LHC announced the dis- arXiv:1210.6553v2 [hep-ph] 15 Feb 2013 covery of the Higgs-like resonance at 126 GeV, [1], and their first measurements indicate intriguing deviations from the Standard Model (SM) expectations. Whether these data signal that a non-minimal Higgs mechanism is indeed at work and if so what it is, are among the hottest questions in particle physics these days. In the past decades, many non-minimal Higgs sectors have been considered, [2]. One conceptually simple and phenomenologically attractive class of models involves several Higgs doublets with identical quantum numbers (N-Higgs-doublet models, NHDM). Its simplest version with only two doublets, 2HDM, was proposed decades ago, [3], but it is still actively studied, see [4] for a recent review, and it has now become a standard reference model of the beyond the Standard Model (bSM) physics. Constructions with more than two doublets are also extensively investigated, [5{14]. Many bSM models aim at providing a natural explanation for the numerical values of (some of) the SM parameters. Often, it is done by invoking additional symmetries in the 1 model. These are not related with the gauge symmetries of the SM but rather reflect extra symmetry structures in the \horizontal space" of the model. One of the main phenomenological motivations in working with several doublets is the ease with which one can introduce various symmetry groups. Indeed, Higgs fields with identical quantum numbers can mix, and it is possible that some of these Higgs-family mixing transformations leave the scalar sector invariant. Even in 2HDM, presence of such a symmetry in the lagrangian and its possible spontaneous violation can lead to a number of remarkable phenomena such as various forms of CP -violation, [3,15], non-standard thermal phase transitions which may be relevant for the early Universe, [16], natural scalar dark matter candidates, [17]. For models with three or more doublets, an extra motivation is the possibility to incorporate into the Higgs sector non- abelian finite symmetry groups, which can then lead to interesting patterns in the fermionic mass matrices (for a general introduction into discrete symmetry groups relevant for particle physics, see [18]). In this respect, the very popular symmetry group has been A4, [7, 8], the smallest finite group with a three-dimensional irreducible representation, but larger symmetry groups also received some attention, [5,9,10]. Given the importance of symmetries for the NHDM phenomenology, it is natural to ask: which symmetry groups can be implemented in the scalar sector of NHDM for a given N? In the two-Higgs-doublet model (2HDM), this question has been answered several years ago, [19, 20], see also [4] for a review. Focusing on discrete symmetries, the only realizable 2 Higgs-family symmetry groups are Z2 and (Z2) . The Z2 group can be generated, for example, by the sign flip of one of the doublets (and it does not matter which, because once we focus on the scalar sector only, the simultaneous sign flip of both doublets does not change the 2 lagrangian), while the (Z2) group is generated by sign flips and the exchange φ1 $ φ2. If 3 generalized-CP transformations are also included, then (Z2) becomes realizable as well, the additional generator being simply the CP conjugation. With more than two doublets, the problem remains open. Although several attempts have been made in past to classify at least some symmetries in NHDM, [11, 12, 14], they led only to very partial results. The main obstacle here was the lack of the completeness criterion. Although many obvious symmetry groups could be immediately guessed, it was not clear how to prove that the given potential does not have other symmetries. An even more difficult problem is to prove that no other symmetry group can be implemented for a given N. In the recent paper [21] we found such a criterion for abelian symmetry groups in NHDM for arbitrary N. Since abelian subgroups are the basic building blocks of any group, classification of realizable abelian symmetry groups in NHDM was an important milestone. We stress that this task is different from just classifying all abelian subgroups of SU(3), because invariance of the Higgs potential places strong and non-trivial restrictions on possible symmetry groups. In this paper, we solve the classification problem for all finite symmetry groups in 3HDM, including non-abelian groups. We do this by using the abelian groups in 3HDM found in [21] and by applying certain results and methods from the theory of solvable groups. Some of these results were already briefly described in [22]. Here, we present a detailed derivation of this classification together with an introduction to the relevant methods from finite group theory. In addition, we extend the analysis to symmetry groups which include both Higgs-family and generalized-CP transformations. This work, therefore, solves the problem of classification of finite reparametrization symmetry groups in 3HDM. We would like to stress one important feature in which our method differs from more traditional approaches to symmetry classification problem, at least within the bSM physics. 2 Usually, one starts by imposing invariance under certain transformations, and then one tries to recognize the symmetry group of the resulting potential. In this way it is very difficult to see whether all possible symmetries are exhausted. We approach the problem the other way around. We first restrict the list of finite groups which can appear as symmetry groups of 3HDM, and then we check one by one whether these groups can indeed be implemented. The structure of this paper is the following. In Section 2 we describe different types of symmetries in the scalar sector of NHDM and discuss the important concept of realizable symmetry groups. Section 3 contains an elementary introduction into the theory of (finite) solvable groups. Although it contains pure mathematics, we put it in the main text because it is a key part of the group-theoretic step of our classification, which is presented in Section 4. Then, in Section 5 we describe the methods which we will use to prove the absence of continu- ous symmetries. Sections 6 and 7 contain the main results of the paper: explicit constructions of the realizable symmetry groups and of the potentials symmetric under each group. Finally, in Section 8 we summarize and discuss our results. For the reader's convenience, we list in the Appendix potentials for each of the realizable non-abelian symmetry groups. 2 Symmetries of the scalar sector of multi-Higgs-doublet models 2.1 Reparametrization transformations In NHDM we introduce N complex Higgs doublets with the electroweak isospin Y = 1=2, which interact with the gauge bosons and matter fields in the standard way, and also self- interact via a Higgs potential. The generic renormalizable Higgs potential can contain only quadratic and quartic gauge-invariant terms, and it can be compactly written as [23,24]: y y y V = Yab(φaφb) + Zabcd(φaφb)(φcφd) ; (1) where all indices run from 1 to N. Coefficients of the potential are grouped into components 2 2 2 of tensors Yab and Zabcd; there are N independent components in Y and N (N + 1)=2 independent components in Z. In this work we focus only on the scalar sector of the NHDM. Therefore, once coefficients Yab and Zabcd are given, the model is completely defined, and one should be able to express all its properties (the number and the positions of extrema, the spectrum and interactions of the physical Higgs bosons) via components of Y 's and Z's. This explicit expression, however, cannot be written via elementary functions, and it remains unknown in the general case for any N > 2. A very important feature of the most general potential is that any non-degenerate lin- ear transformation in the space of Higgs doublets belonging to the group GL(2; C) keeps the generic form of the potential, changing only the coefficients of Y and Z. We call such a trans- formation a Higgs-basis change. In addition, the CP transformation, which maps doublets y to their hermitean conjugates φa ! φa, also keeps the generic form of the potential, up to coefficient modification. Its combination with a Higgs-basis change represents a transforma- tion which is usually called a generalized-CP transformation, [25].